Sổ tay kết cấu thép - Section 13

TRUSS BRIDGES

13.1SECTION 13

TRUSS BRIDGES*

John M Kulicki, P.E.

President and Chief Engineer

Joseph E Prickett, P.E.

Senior Associate

David H LeRoy, P.E.

Vice President

Modjeski and Masters, Inc., Harrisburg, Pennsylvania

A truss is a structure that acts like a beam but with major components, or members, subjectedprimarily to axial stresses The members are arranged in triangular patterns Ideally, the endof each member at a joint is free to rotate independently of the other members at the joint.If this does not occur, secondary stresses are induced in the members Also if loads occurother than at panel points, or joints, bending stresses are produced in the members.

Though trusses were used by the ancient Romans, the modern truss concept seems tohave been originated by Andrea Palladio, a sixteenth century Italian architect From his timeto the present, truss bridges have taken many forms.

Early trusses might be considered variations of an arch They applied horizontal thrustsat the abutments, as well as vertical reactions, In 1820, Ithiel Town patented a truss that canbe considered the forerunner of the modern truss Under vertical loading, the Town trussexerted only vertical forces at the abutments But unlike modern trusses, the diagonals, orweb systems, were of wood lattice construction and chords were composed of two or moretimber planks.

In 1830, Colonel Long of the U.S Corps of Engineers patented a wood truss with asimpler web system In each panel, the diagonals formed an X The next major step camein 1840, when William Howe patented a truss in which he used wrought-iron tie rods forvertical web members, with X wood diagonals This was followed by the patenting in 1844of the Pratt truss with wrought-iron X diagonals and timber verticals.

The Howe and Pratt trusses were the immediate forerunners of numerous iron bridges.In a book published in 1847, Squire Whipple pointed out the logic of using cast iron incompression and wrought iron in tension He constructed bowstring trusses with cast-ironverticals and wrought-iron X diagonals.

*Revised and updated from Sec 12, ‘‘Truss Bridges,’’ by Jack P Shedd, in the first edition.

These trusses were statically indeterminate Stress analysis was difficult Latter, simplerweb systems were adopted, thus eliminating the need for tedious and exacting design pro-cedures.

To eliminate secondary stresses due to rigid joints, early American engineers constructedpin-connected trusses European engineers primarily used rigid joints Properly proportioned,the rigid trusses gave satisfactory service and eliminated the possibility of frozen pins, whichinduce stresses not usually considered in design Experience indicated that rigid and pin-connected trusses were nearly equal in cost, except for long spans Hence, modern designfavors rigid joints.

Many early truss designs were entirely functional, with little consideration given to pearance Truss members and other components seemed to lie in all possible directions andto have a variety of sizes, thus giving the impression of complete disorder Yet, appearanceof a bridge often can be improved with very little increase in construction cost By the1970s, many speculated that the cable-stayed bridge would entirely supplant the truss, excepton railroads But improved design techniques, including load-factor design, and streamlineddetailing have kept the truss viable For example, some designs utilize Warren trusses withoutverticals In some cases, sway frames are eliminated and truss-type portals are replaced withbeam portals, resulting in an open appearance.

ap-Because of the large number of older trusses still in the transportation system, somehistorical information in this section applies to those older bridges in an evaluation or re-habilitation context.

(H J Hopkins, ‘‘A Span of Bridges,’’ Praeger Publishers, New York; S P Timoshenko,‘‘History of Strength of Materials,’’ McGraw-Hill Book Company, New York).

The design of truss bridges usually follows the specifications of the American Associationof State Highway and Transportation Officials (AASHTO) or the Manual of the AmericanRailway Engineering and Maintenance of Way Association (AREMA) (Sec 10) A transitionin AASHTO specifications is currently being made from the ‘‘Standard Specifications forHighway Bridges,’’ Sixteenth Edition, to the ‘‘LRFD Specifications for Highway Bridges,’’Second Edition The ‘‘Standard Specification’’ covers service load design of truss bridges,and in addition, the ‘‘Guide Specification for the Strength Design of Truss Bridges,’’ coversextension of the load factor design process permitted for girder bridges in the ‘‘StandardSpecifications’’ to truss bridges Where the ‘‘Guide Specification’’ is silent, applicable pro-visions of the ‘‘Standard Specification’’ apply.

To clearly identify which of the three AASHTO specifications are being referred to inthis section, the following system will be adopted If the provision under discussion appliesto all the specifications, reference will simply be made to the ‘‘AASHTO Specifications’’.Otherwise, the following notation will be observed:

‘‘AASHTO SLD Specifications’’ refers to the service load provisions of ‘‘Standard ifications for Highway Bridges’’

Spec-‘‘AASHTO LFD Specifications’’ refers to ‘‘Guide Specification for the Strength Designof Truss Bridges’’

‘‘AASHTO LRFD Specifications’’ refers to ‘‘LRFD Specifications for Highway Bridges.’’

Principal parts of a highway truss bridge are indicated in Fig 13.1; those of a railroad trussare shown in Fig 13.2.

FIGURE 13.1 Cross section shows principal parts of a deck-truss highway bridge.

Joints are intersections of truss members Joints along upper and lower chords often are

referred to as panel points To minimize bending stresses in truss members, live loads erally are transmitted through floor framing to the panel points of either chord in older,shorter-span trusses Bending stresses in members due to their own weight was often ignoredin the past In modern trusses, bending due to the weight of the members should be consid-ered.

gen-Chords are top and bottom members that act like the flanges of a beam They resist the

tensile and compressive forces induced by bending In a constant-depth truss, chords areessentially parallel They may, however, range in profile from nearly horizontal in a mod-erately variable-depth truss to nearly parabolic in a bowstring truss Variable depth oftenimproves economy by reducing stresses where chords are more highly loaded, around mid-span in simple-span trusses and in the vicinity of the supports in continuous trusses.

Web members consist of diagonals and also often of verticals Where the chords are

essentially parallel, diagonals provide the required shear capacity Verticals carry shear, vide additional panel points for introduction of loads, and reduce the span of the chordsunder dead-load bending When subjected to compression, verticals often are called posts,and when subjected to tension, hangers Usually, deck loads are transmitted to the trussesthrough end connections of floorbeams to the verticals.

pro-Counters, which are found on many older truss bridges still in service, are a pair of

diagonals placed in a truss panel, in the form of an X, where a single diagonal would be

FIGURE 13.2 Cross section shows principal parts of a through-truss railway bridge.

subjected to stress reversals Counters were common in the past in short-span trusses Suchshort-span trusses are no longer economical and have been virtually totally supplanted bybeam and girder spans X pairs are still used in lateral trusses, sway frames and portals, butare seldom designed to act as true counters, on the assumption that only one counter acts ata time and carries the maximum panel shear in tension This implies that the companioncounter takes little load because it buckles In modern design, counters are seldom used inthe primary trusses Even in lateral trusses, sway frames, and portals, X-shaped trusses areusually comprised of rigid members, that is, members that will not buckle If adjustablecounters are used, only one may be placed in each truss panel, and it should have openturnbuckles AASHTO LRFD specifies that counters should be avoided The commentary tothat provision contains reference to the historical initial force requirement of 10 kips Designof such members by AASHTO SLD or LFD Specifications should include an allowance of10 kips for initial stress Sleeve nuts and loop bars should not be used.

End posts are compression members at supports of simple-span tusses Wherever

prac-tical, trusses should have inclined end posts Laterally unsupported hip joints should not beused.

Working lines are straight lines between intersections of truss members To avoid bending

stresses due to eccentricity, the gravity axes of truss members should lie on working lines.Some eccentricity may be permitted, however, to counteract dead-load bending stresses.Furthermore, at joints, gravity axes should intersect at a point If an eccentric connection isunavoidable, the additional bending caused by the eccentricity should be included in thedesign of the members utilizing appropriate interaction equations.

AASHTO Specifications require that members be symmetrical about the central plane ofa truss They should be proportioned so that the gravity axis of each section lies as nearlyas practicable in its center.

Connections may be made with welds or high-strength bolts AREMA practice, however,excludes field welding, except for minor connections that do not support live load.

The deck is the structural element providing direct support for vehicular loads Where

the deck is located near the bottom chords (through spans), it should be supported by onlytwo trusses.

Floorbeams should be set normal or transverse to the direction of traffic They and their

connections should be designed to transmit the deck loads to the trusses.

Stringers are longitudinal beams, set parallel to the direction of traffic They are used to

transmit the deck loads to the floorbeams If stringers are not used, the deck must be designedto transmit vehicular loads to the floorbeams.

Lateral bracing should extend between top chords and between bottom chords of the

two trusses This bracing normally consists of trusses placed in the planes of the chords toprovide stability and lateral resistance to wind Trusses should be spaced sufficiently far apartto preclude overturning by design lateral forces.

Sway bracing may be inserted between truss verticals to provide lateral resistance in

vertical planes Where the deck is located near the bottom chords, such bracing, placedbetween truss tops, must be kept shallow enough to provide adequate clearance for passageof traffic below it Where the deck is located near the top chords, sway bracing should extendin full-depth of the trusses.

Portal bracing is sway bracing placed in the plane of end posts In addition to serving

the normal function of sway bracing, portal bracing also transmits loads in the top lateralbracing to the end posts (Art 13.6).

Skewed bridges are structures supported on piers that are not perpendicular to the planes

of the trusses The skew angle is the angle between the transverse centerline of bearings

and a line perpendicular to the longitudinal centerline of the bridge.

13.3TYPES OF TRUSSES

Figure 13.3 shows some of the common trusses used for bridges Pratt trusses have

diag-onals sloping downward toward the center and parallel chords (Fig 13.3a) Warren trusses,

FIGURE 13.3 Types of simple-span truss bridges.

with parallel chords and alternating nals, are generally, but not always, con-

diago-structed with verticals (Fig 13.3c) to reduce

panel size When rigid joints are used, suchtrusses are favored because they provide anefficient web system Most modern bridgesare of some type of Warren configuration.

Parker trusses (Fig 13.3d ) resemble

Pratt trusses but have variable depth As inother types of trusses, the chords provide acouple that resists bending moment Withlong spans, economy is improved by creatingthe required couple with less force by spac-ing the chords farther apart The Parker truss,when simply supported, is designed to haveits greatest depth at midspan, where momentis a maximum For greatest chord economy,the top-chord profile should approximate aparabola Such a curve, however, providestoo great a change in slope of diagonals, withsome loss of economy in weights of diago-nals In practice, therefore, the top-chordprofile should be set for the greatest changein truss depth commensurate with reasonablediagonal slopes; for example, between 40⬚

and 60⬚with the horizontal.

K trusses (Fig 13.3e) permit deep

trusses with short panels to have diagonalswith acceptable slopes Two diagonals generally are placed in each panel to intersect atmidheight of a vertical Thus, for each diagonal, the slope is half as large as it would be ifa single diagonal were used in the panel The short panels keep down the cost of the floorsystem This cost would rise rapidly if panel width were to increase considerably withincrease in span Thus, K trusses may be economical for long spans, for which deep trussesand narrow panels are desirable These trusses may have constant or variable depth.

Bridges also are classified as highway or railroad, depending on the type of loading thebridge is to carry Because highway loading is much lighter than railroad, highway trussesgenerally are built of much lighter sections Usually, highways are wider than railways, thusrequiring wider spacing of trusses.

Trusses are also classified as to location of deck: deck, through, or half-through trusses.

Deck trusses locate the deck near the top chord so that vehicles are carried above the chord.Through trusses place the deck near the bottom chord so that vehicles pass between the

trusses Half-through trusses carry the deck so high above the bottom chord that lateral and

sway bracing cannot be placed between the top chords The choice of deck or throughconstruction normally is dictated by the economics of approach construction.

The absence of top bracing in half-through trusses calls for special provisions to resistlateral forces AASHTO Specifications require that truss verticals, floorbeams, and their endconnections be proportioned to resist a lateral force of at least 0.30 kip per lin ft, applied atthe top chord panel points of each truss The top chord of a half-through truss should bedesigned as a column with elastic lateral supports at panel points The critical buckling forceof the column, so determined, should be at least 50% larger than the maximum force inducedin any panel of the top chord by dead and live loads plus impact Thus, the verticals haveto be designed as cantilevers, with a concentrated load at top-chord level and rigid connectionto a floorbeam This system offers elastic restraint to buckling of the top chord The analysisof elastically restrained compression members is covered in T V Galambos, ‘‘Guide toStability Design Criteria for Metal Structures,’’ Structural Stability Research Council.

13.4BRIDGE LAYOUT

Trusses, offering relatively large depth, open-web construction, and members subjected marily to axial stress, provide large carrying capacity for comparatively small amounts ofsteel For maximum economy in truss design, the area of metal furnished for members shouldbe varied as often as required by the loads To accomplish this, designers usually have tospecify built-up sections that require considerable fabrication, which tend to offset some ofthe savings in steel.

pri-Truss Spans. Truss bridges are generally comparatively easy to erect, because light ment often can be used Assembly of mechanically fastened joints in the field is relativelylabor-intensive, which may also offset some of the savings in steel Consequently, trussesseldom can be economical for highway bridges with spans less than about 450 ft.

equip-Railroad bridges, however, involve different factors, because of the heavier loading.Trusses generally are economical for railroad bridges with spans greater than 150 ft.

The current practical limit for simple-span trusses is about 800 ft for highway bridgesand about 750 ft for railroad bridges Some extension of these limits should be possible withimprovements in materials and analysis, but as span requirements increase, cantilever orcontinuous trusses are more efficient The North American span record for cantilever con-struction is 1,600 ft for highway bridges and 1,800 ft for railroad bridges.

For a bridge with several truss spans, the most economical pier spacing can be determinedafter preliminary designs have been completed for both substructure and superstructure Oneguideline provides that the cost of one pier should equal the cost of one superstructure span,excluding the floor system In trial calculations, the number of piers initially assumed maybe increased or decreased by one, decreasing or increasing the truss spans Cost of trussspans rises rapidly with increase in span A few trial calculations should yield a satisfactorypicture of the economics of the bridge layout Such an analysis, however, is more suitablefor approach spans than for main spans In most cases, the navigation or hydraulic require-ment is apt to unbalance costs in the direction of increased superstructure cost Furthermore,girder construction is currently used for span lengths that would have required approachtrusses in the past.

Panel Dimensions. To start economic studies, it is necessary to arrive at economic portions of trusses so that fair comparisons can be made among alternatives Panel lengthswill be influenced by type of truss being designed They should permit slope of the diagonalsbetween 40⬚ and 60⬚ with the horizontal for economic design If panels become too long,the cost of the floor system substantially increases and heavier dead loads are transmitted tothe trusses A subdivided truss becomes more economical under these conditions.

pro-For simple-span trusses, experience has shown that a depth-span ratio of 1:5 to 1:8 yieldseconomical designs Some design specifications limit this ratio, with 1:10 a common histor-ical limit For continuous trusses with reasonable balance of spans, a depth-span ratio of1:12 should be satisfactory Because of the lighter live loads for highways, somewhat shal-lower depths of trusses may be used for highway bridges than for railway bridges.

Designers, however, do not have complete freedom in selection of truss depth Certainphysical limitations may dictate the depth to be used For through-truss highway bridges,for example, it is impractical to provide a depth of less than 24 ft, because of the necessityof including suitable sway frames Similarly, for through railway trusses, a depth of at least30 ft is required The trend toward double-stack cars encourages even greater minimumdepths.

Once a starting depth and panel spacing have been determined, permutation of primarygeometric variables can be studied efficiently by computer-aided design methods In fact,preliminary studies have been carried out in which every primary truss member is designed

for each choice of depth and panel spacing, resulting in a very accurate choice of thoseparameters.

Bridge Cross Sections. Selection of a proper bridge cross section is an important mination by designers In spite of the large number of varying cross sections observed intruss bridges, actual selection of a cross section for a given site is not a large task Forinstance, if a through highway truss were to be designed, the roadway width would determinethe transverse spacing of trusses The span and consequent economical depth of trusses woulddetermine the floorbeam spacing, because the floorbeams are located at the panel points.Selection of the number of stringers and decisions as to whether to make the stringers simplespans between floorbeams or continuous over the floorbeams, and whether the stringers andfloorbeams should be composite with the deck, complete the determination of the crosssection.

deter-Good design of framing of floor system members requires attention to details In the past,

many points of stress relief were provided in floor systems Due to corrosion and wear

resulting from use of these points of movement, however, experience with them has notalways been good Additionally, the relative movement that tends to occur between the deckand the trusses may lead to out-of-plane bending of floor system members and possiblefatigue damage Hence, modern detailing practice strives to eliminate small unconnectedgaps between stiffeners and plates, rapid change in stiffness due to excessive flange coping,and other distortion fatigue sites Ideally, the whole structure is made to act as a unit, thuseliminating distortion fatigue.

Deck trusses for highway bridges present a few more variables in selection of crosssection Decisions have to be made regarding the transverse spacing of trusses and whetherthe top chords of the trusses should provide direct support for the deck Transverse spacingof the trusses has to be large enough to provide lateral stability for the structure Narrowertruss spacings, however, permit smaller piers, which will help the overall economy of thebridge.

Cross sections of railway bridges are similarly determined by physical requirements ofthe bridge site Deck trusses are less common for railway bridges because of the extra lengthof approach grades often needed to reach the elevation of the deck Also, use of throughtrusses offers an advantage if open-deck construction is to be used With through-trusses,only the lower chords are vulnerable to corrosion caused by salt and debris passing throughthe deck.

After preliminary selection of truss type, depth, panel lengths, member sizes, lateral tems, and other bracing, designers should review the appearance of the entire bridge Es-thetics can often be improved with little economic penalty.

sys-13.5DECK DESIGN

For most truss members, the percentage of total stress attributable to dead load increases asspan increases Because trusses are normally used for long spans, and a sizable portion ofthe dead load (particularly on highway bridges) comes from the weight of the deck, a light-weight deck is advantageous It should be no thicker than actually required to support thedesign loading.

In the preliminary study of a truss, consideration should be given to the cost, durability,maintainability, inspectability, and replaceability of various deck systems, including trans-verse, longitudinal, and four-way reinforced concrete decks, orthotropic-plate decks, andconcrete-filled or overlaid steel grids Open-grid deck floors will seldom be acceptable fornew fixed truss bridges but may be advantageous in rehabilitation of bridges and for movablebridges.

The design procedure for railroad bridge decks is almost entirely dictated by the proposedcross section Designers usually have little leeway with the deck, because they are requiredto use standard railroad deck details wherever possible.

Deck design for a highway bridge is somewhat more flexible Most highway bridges havea reinforced-concrete slab deck, with or without an asphalt wearing surface Reinforcedconcrete decks may be transverse, longitudinal or four-way slabs.

• Transverse slabs are supported on stringers spaced close enough so that all the bending inthe slabs is in a transverse direction.

• Longitudinal slabs are carried by floorbeams spaced close enough so that all the bendingin the slabs is in a longitudinal direction Longitudinal concrete slabs are practical forshort-span trusses where floorbeam spacing does not exceed about 20 ft For larger spacing,the slab thickness becomes so large that the resultant dead load leads to an uneconomictruss design Hence, longitudinal slabs are seldom used for modern trusses.

• Four-way slabs are supported directly on longitudinal stringers and transverse floorbeams.Reinforcement is placed in both directions The most economical design has a spacing ofstringers about equal to the spacing of floorbeams This restricts use of this type of floorsystem to trusses with floorbeam spacing of about 20 ft As for floor systems with alongitudinal slab, four-way slabs are generally uneconomical for modern bridges.

13.6LATERAL BRACING, PORTALS, AND SWAY FRAMES

Lateral bracing should be designed to resist the following: (1) Lateral forces due to windpressure on the exposed surface of the truss and on the vertical projection of the live load.(2) Seismic forces, (3) Lateral forces due to centrifugal forces when the track or roadway iscurved (4) For railroad bridges, lateral forces due to the nosing action of locomotives causedby unbalanced conditions in the mechanism and also forces due to the lurching movementof cars against the rails because of the play between wheels and rails Adequate bracing isone of the most important requirements for a good design.

Since the loadings given in design specifications only approximate actual loadings, itfollows that refined assumptions are not warranted for calculation of panel loads on lateraltrusses The lateral forces may be applied to the windward truss only and divided betweenthe top and bottom chords according to the area tributary to each A lateral bracing truss isplaced between the top chords or the bottom chords, or both, of a pair of trusses to carrythese forces to the ends of the trusses.

Besides its use to resist lateral forces, other purposes of lateral bracing are to providestability, stiffen structures and prevent unwarranted lateral vibration In deck-truss bridges,however, the floor system is much stiffer than the lateral bracing Here, the major purposeof lateral bracing is to true-up the bridges and to resist wind load during erection.

The portal usually is a sway frame extending between a pair of trusses whose purposealso is to transfer the reactions from a lateral-bracing truss to the end posts of the trusses,and, thus, to the foundation This action depends on the ability of the frame to resist trans-verse forces.

The portal is normally a statically indeterminate frame Because the design loadings areapproximate, an exact analysis is seldom warranted It is normally satisfactory to makesimplifying assumptions For example, a plane of contraflexure may be assumed halfwaybetween the bottom of the portal knee brace and the bottom of the post The shear on theplane may be assumed divided equally between the two end posts.

Sway frames are placed between trusses, usually in vertical planes, to stiffen the structure(Fig 13.1 and 13.2) They should extend the full depth of deck trusses and should be madeas deep as possible in through trusses The AASHTO SLD Specifications require sway frames

in every panel But many bridges are serving successfully with sway frames in every otherpanel, even lift bridges whose alignment is critical Some designs even eliminate sway framesentirely The AASHTO LRFD Specifications makes the use and number of sway frames amatter of design concept as expressed in the analysis of the structural system.

Diagonals of sway frames should be proportioned for slenderness ratio as compressionmembers With an X system of bracing, any shear load may be divided equally between thediagonals An approximate check of possible loads in the sway frame should be made toensure that stresses are within allowable limits.

13.7RESISTANCE TO LONGITUDINAL FORCES

Acceleration and braking of vehicular loads, and longitudinal wind, apply longitudinal loadsto bridges In highway bridges, the magnitudes of these forces are generally small enoughthat the design of main truss members is not affected In railroad bridges, however, chordsthat support the floor system might have to be increased in section to resist tractive forces.In all truss bridges, longitudinal forces are of importance in design of truss bearings andpiers.

In railway bridges, longitudinal forces resulting from accelerating and braking may inducesevere bending stresses in the flanges of floorbeams, at right angles to the plane of the web,unless such forces are diverted to the main trusses by traction frames In single-track bridges,a transverse strut may be provided between the points where the main truss laterals cross

the stringers and are connected to them (Fig 13.4a) In double-track bridges, it may benecessary to add a traction truss (Fig 13.4b).

When the floorbeams in a double-track bridge are so deep that the bottoms of the stringersare a considerable distance above the bottoms of the floorbeams, it may be necessary to raisethe plane of the main truss laterals from the bottom of the floorbeams to the bottom of thestringers If this cannot be done, a complete and separate traction frame may be providedeither in the plane of the tops of the stringers or in the plane of their bottom flanges.

The forces for which the traction frames are designed are applied along the stringers Themagnitudes of these forces are determined by the number of panels of tractive or brakingforce that are resisted by the frames When one frame is designed to provide for severalpanels, the forces may become large, resulting in uneconomical members and connections.

13.8TRUSS DESIGN PROCEDURE

The following sequence may serve as a guide to the design of truss bridges:

• Select span and general proportions of the bridge, including a tentative cross section.• Design the roadway or deck, including stringers and floorbeams.

• Design upper and lower lateral systems.• Design portals and sway frames.

• Design posts and hangers that carry little stress or loads that can be computed without acomplete stress analysis of the entire truss.

• Compute preliminary moments, shears, and stresses in the truss members.• Design the upper-chord members, starting with the most heavily stressed member.• Design the lower-chord members.

• Design the web members.

FIGURE 13.4 Lateral bracing and traction trusses for resisting longitudinalforces on a truss bridge.

• Recalculate the dead load of the truss and compute final moments and stresses in trussmembers.

• Design joints, connections, and details.• Compute dead-load and live-load deflections.

• Check secondary stresses in members carrying direct loads and loads due to wind.• Review design for structural integrity, esthetics, erection, and future maintenance and in-

spection requirements.

13.8.1Analysis for Vertical Loads

Determination of member forces using conventional analysis based on frictionless joints isoften adequate when the following conditions are met:

1 The plane of each truss of a bridge, the planes through the top chords, and the planes

through the bottom chords are fully triangulated.

2 The working lines of intersecting truss members meet at a point.

3 Cross frames and other bracing prevent significant distortions of the box shape formed

by the planes of the truss described above.

4 Lateral and other bracing members are not cambered; i.e., their lengths are based on the

final dead-load position of the truss.

5 Primary members are cambered by making them either short or long by amounts equal

to, and opposite in sign to, the axial compression or extension, respectively, resultingfrom dead-load stress Camber for trusses can be considered as a correction for dead-loaddeflection (If the original design provided excess vertical clearance and the engineers didnot object to the sag, then trusses could be constructed without camber Most people,however, object to sag in bridges.) The cambering of the members results in the trussbeing out of vertical alignment until all the dead loads are applied to the structure (geo-metric condition).

When the preceding conditions are met and are rigorously modeled, three-dimensionalcomputer analysis yields about the same dead-load axial forces in the members as the con-ventional pin-connected analogy and small secondary moments resulting from the self-weightbending of the member Application of loads other than those constituting the geometriccondition, such as live load and wind, will result in sag due to stressing of both primary andsecondary members in the truss.

Rigorous three-dimensional analysis has shown that virtually all the bracing membersparticipate in live-load stresses As a result, total stresses in the primary members are reducedbelow those calculated by the conventional two-dimensional pin-connected truss analogy.Since trusses are usually used on relatively long-span structures, the dead-load stress con-stitutes a very large part of the total stress in many of the truss members Hence, the savingsfrom use of three-dimensional analysis of the live-load effects will usually be relatively small.This holds particularly for through trusses where the eccentricity of the live load, and, there-fore, forces distributed in the truss by torsion are smaller than for deck trusses.

The largest secondary stresses are those due to moments produced in the members by theresistance of the joints to rotation Thus, the secondary stresses in a pin-connected truss aretheoretically less significant than those in a truss with mechanically fastened or welded joints.In practice, however, pinned joints always offer frictional resistance to rotation, even whennew If pin-connected joints freeze because of dirt, or rust, secondary stresses might becomehigher than those in a truss with rigid connections Three-dimensional analysis will however,quantify secondary stresses, if joints and framing of members are accurately modeled If thesecondary stress exceeds 4 ksi for tension members or 3 ksi for compression members, boththe AASHTO SLD and LFD Specifications require that excess be treated as a primary stress.The AASHTO LRFD Specifications take a different approach including:

• A requirement to detail the truss so as to make secondary force effects as small as practical.• A requirement to include the bending caused by member self-weight, as well as moments

resulting from eccentricities of joint or working lines.

• Relief from including both secondary force effects from joint rotation and floorbeam flection if the component being designed is more than ten times as long as it is wide inthe plane of bending.

de-When the working lines through the centroids of intersecting members do not intersectat the joint, or where sway frames and portals are eliminated for economic or esthetic pur-poses, the state of bending in the truss members, as well as the rigidity of the entire system,should be evaluated by a more rigorous analysis than the conventional.

The attachment of floorbeams to truss verticals produces out-of-plane stresses, whichshould be investigated in highway bridges and must be accounted for in railroad bridges,due to the relatively heavier live load in that type of bridge An analysis of a frame composedof a floorbeam and all the truss members present in the cross section containing the floorbeam is usually adequate to quantify this effect.

Deflection of trusses occurs whenever there are changes in length of the truss members.These changes may be due to strains resulting from loads on the truss, temperature variations,or fabrication effects or errors Methods of computing deflections are similar in all threecases Prior to the introduction of computers, calculation of deflections in trusses was alaborious procedure and was usually determined by energy or virtual work methods or bygraphical or semigraphical methods, such as the Williot-Mohr diagram With the widespreadavailability of matrix structural analysis packages, the calculation of deflections and analysisof indeterminant trusses are speedily executed.

(See also Arts 3.30, 3.31, and 3.34 to 3.39).

13.8.2Analysis for Wind Loads

The areas of trusses exposed to wind normal to their longitudinal axis are computed bymultiplying widths of members as seen in elevation by the lengths center to center of inter-sections The overlapping areas at intersections are assumed to provide enough surplus toallow for the added areas of gussets The AREMA Manual specifies that for railway bridgesthis truss area be multiplied by the number of trusses, on the assumption that the wind strikeseach truss fully (except where the leeward trusses are shielded by the floor system) TheAASHTO Specifications require that the area of the trusses and floor as seen in elevation bemultiplied by a wind pressure that accounts for 11⁄2times this area being loaded by wind.

The area of the floor should be taken as that seen in elevation, including stringers, deck,railing, and railing pickets.

AREMA specifies that when there is no live load on the structure, the wind pressureshould be taken as at least 50 psf, which is equivalent to a wind velocity of about 125 mph.When live load is on the structure, reduced wind pressures are specified for the trusses plusfull wind load on the live load: 30 psf on the bridge, which is equivalent to a 97-mph wind,and 300 lb per lin ft on the live load on one track applied 8 ft above the top of rail.

AASHTO SLD Specifications require a wind pressure on the structure of 75 psf Totalforce, lb per lin ft, in the plane of the windward chords should be taken as at least 300 andin the plane of the leeward chords, at least 150 When live load is on the structure, thesewind pressures can be reduced 70% and combined with a wind force of 100 lb per lin ft onthe live load applied 6 ft above the roadway The AASHTO LFD Specifications do notexpressly address wind loads, so the SLD Specifications pertain by default.

Article 3.8 of the AASHTO LRFD Specifications establish wind loads consistent withthe format and presentation currently used in meteorology Wind pressures are related to a

base wind velocity, VB, of 100 mph as common in past specifications If no better information

is available, the wind velocity at 30 ft above the ground, V30, may be taken as equal to the

base wind, VB The height of 30 ft was selected to exclude ground effects in open terrain.Alternatively, the base wind speed may be taken from Basic Wind Speed Charts available

in the literature, or site specific wind surveys may be used to establish V30.

At heights above 30 ft, the design wind velocity, VDZ, mph, on a structure at a height, Z,

ft, may be calculated based on characteristic meteorology quantities related to the terrain

over which the winds approach as follows Select the friction velocity, V0, and friction length,

Z0, from Table 13.1 Then calculate the velocity from

For design, Table 13.2 gives the base pressure, PB, ksf, acting on various structural

com-ponents for a base wind velocity of 100 mph The design wind pressure, PD, ksf, for the

design wind velocity, V , mph, is calculated from

TABLE 13.1 Basic Wind ParametersTerrainOpen

V0, mph8.2010.912.0

TABLE 13.2 Base Pressures, PBfor Base Wind

Velocity, VB, of 100 mphStructural

Windwardload, ksf

Leewardload, ksfTrusses, Columns,

and Arches

Large FlatSurfaces

AASHTO Specifications also require that wind pressure be applied to vehicular live load.

Wind Analysis. Wind analysis is typically carried out with the aid of computers with aspace truss and some frame members as a model It is helpful, and instructive, to employ asimplified, noncomputer method of analysis to compare with the computer solution to exposemajor modeling errors that are possible with space models Such a simplified method ispresented in the following.

Idealized Wind-Stress Analysis of a through Truss with Inclined End Posts. The windloads computed as indicated above are applied as concentrated loads at the panel points.

A through truss with parallel chords may be considered as having reactions to the toplateral bracing system only at the main portals The effect of intermediate sway frames,therefore, is ignored The analysis is applied to the bracing and to the truss members.

The lateral bracing members in each panel are designed for the maximum shear in thepanel resulting from treating the wind load as a moving load; that is, as many panels areloaded as necessary to produce maximum shear in that panel In design of the top-chordbracing members, the wind load, without live load, usually governs The span for top-chordbracing is from hip joint to hip joint For the bottom-chord members, the reduced windpressure usually governs because of the considerable additional force that usually resultsfrom wind on the live load.

For large trusses, wind stress in the trusses should be computed for both the maximumwind pressure without live load and for the reduced wind pressure with live load and fullwind on the live load Because wind on the live load introduces an effect of ‘‘transfer,’’ as

FIGURE 13.6 Wind on a cantilever truss with curved topchord is resisted by the top lateral system.

described later, the following discussion is for the more general case of a truss with thereduced wind pressure on the structure and with wind on the live load applied 8 ft abovethe top of rail, or 6 ft above the deck.

The effect of wind on the trusses may be considered to consist of three additive parts:• Chord stresses in the fully loaded top and bottom lateral trusses.

• Horizontal component, which is a uniform force of tension in one truss bottom chord

and compression in the other bottom chord, resulting from transfer of the top lateral endreactions down the end portals This may be taken as the top lateral end reaction timesthe horizontal distance from the hip joint to the point of contraflexure divided by thespacing between main trusses It is often conservatively assumed that this point of contra-flexure is at the end of span, and, thus, the top lateral end reaction is multiplied by thepanel length, divided by the spacing between main trusses Note that this convenient as-sumption does not apply to the design of portals themselves.

• Transfer stresses created by the moment of wind on the live load and wind on the floor.

This moment is taken about the plane of the bottom lateral system The wind force on liveload and wind force on the floor in a panel length is multiplied by the height of applicationabove the bracing plane and divided by the distance center to center of trusses to arriveat a total vertical panel load This load is applied downward at each panel point of theleeward truss and upward at each panel point of the windward truss The resulting stressesin the main vertical trusses are then computed.

The total wind stress in any main truss member is arrived at by adding all three effects:chord stresses in the lateral systems, horizontal component, and transfer stresses.

FIGURE 13.5 Top chord in a horizontal plane proximates a curved top chord.

ap-Although this discussion applies to a allel-chord truss, the same method may beapplied with only slight error to a truss withcurved top chord by considering the topchord to lie in a horizontal plane between hipjoints, as shown in Fig 13.5 The nature ofthis error will be described in the following.

par-Wind Stress Analysis of Curved-Chord Cantilever Truss. The additional effects that shouldbe considered in curved-chord trusses are those of the vertical components of the inclinedbracing members These effects may be illustrated by the behavior of a typical cantileverbridge, several panels of which are shown in Fig 13.6.

As transverse forces are applied to the curved top lateral system, the transverse shearcreates stresses in the top lateral bracing members The longitudinal and vertical componentsof these bracing stresses create wind stresses in the top chords and other members of themain trusses The effects of these numerous components of the lateral members may bedetermined by the following simple method:

• Apply the lateral panel loads to the horizontal projection of the top-chord lateral systemand compute all horizontal components of the chord stresses The stresses in the inclined

chords may readily be computed from these horizontal components.

• Determine at every point of slope change in the top chord all the vertical forces acting onthe point from both bracing diagonals and bracing chords Compute the truss stresses inthe vertical main trusses from those forces.

• The final truss stresses are the sum of the two contributions above and also of any transferstress, and of any horizontal component delivered by the portals to the bottom chords.

13.8.3Computer Determination of Wind Stresses

For computer analysis, the structural model is a three-dimensional framework composed ofall the load-carrying members Floorbeams are included if they are part of the bracing systemor are essential for the stability of the structural model.

All wind-load concentrations are applied to the framework at braced points Because thewind loads on the floor system and on the live load do not lie in a plane of bracing, theseloads must be ‘‘transferred’’ to a plane of bracing The accompanying vertical required forequilibrium also should be applied to the framework.

Inasmuch as significant wind moments are produced in open-framed portal members ofthe truss, flexural rigidity of the main-truss members in the portal is essential for stability.Unless the other framework members are released for moment, the computer analysis willreport small moments in most members of the truss.

With cantilever trusses, it is a common practice to analyze the suspended span by itselfand then apply the reactions to a second analysis of the anchor and cantilever arms.

Some consideration of the rotational stiffness of piers about their vertical axis is warrantedfor those piers that support bearings that are fixed against longitudinal translation Such pierswill be subjected to a moment resulting from the longitudinal forces induced by lateral loads.If the stiffness (or flexibility) of the piers is not taken into account, the sense and magnitudeof chord forces may be incorrectly determined.

13.8.4Wind-Induced Vibration of Truss Members

When a steady wind passes by an obstruction, the pressure gradient along the obstructioncauses eddies or vortices to form in the wind stream These occur at stagnation points locatedon opposite sides of the obstruction As a vortex grows, it eventually reaches a size thatcannot be tolerated by the wind stream and is torn loose and carried along in the windstream The vortex at the opposite stagnation point then grows until it is shed The result isa pattern of essentially equally spaced (for small distances downwind of the obstruction) andalternating vortices called the ‘‘Vortex Street’’ or ‘‘von Karman Trail.’’ This vortex street isindicative of a pulsating periodic pressure change applied to the obstruction The frequencyof the vortex shedding and, hence, the frequency of the pulsating pressure, is given by

where V is the wind speed, fps, D is a characteristic dimension, ft, and S is the Strouhal

number, the ratio of velocity of vibration of the obstruction to the wind velocity (Table 13.3).When the obstruction is a member of a truss, self-exciting oscillations of the member inthe direction perpendicular to the wind stream may result when the frequency of vortexshedding coincides with a natural frequency of the member Thus, determination of thetorsional frequency and bending frequency in the plane perpendicular to the wind and sub-stitution of those frequencies into Eq (13.3) leads to an estimate of wind speeds at whichresonance may occur Such vibration has led to fatigue cracking of some truss and archmembers, particularly cable hangers and I-shaped members The preceding proposed use ofEq (13.3) is oriented toward guiding designers in providing sufficient stiffness to reasonably

TABLE 13.3 Strouhal Number for Various Sections*Wind

Formulation for Vibration Analysis,’’ ASCE Journal of Engineering Mechanics, vol 115, no.

TABLE 13.4 Eigenvalue knL and Effective Length Factor K

0.3330.2920.2590.400where ƒn⫽natural frequency of member for each mode corresponding to n⫽1, 2, 3,

knL⫽eigenvalue for each mode (see Table 13.4)

K⫽effective length factor (see Table 13.4)

L⫽length of the member, in

I⫽moment of inertia, in4, of the member cross section

a⫽coefficient dependent on the physical properties of the member

⫽兹EIg /␥Afor bending

E⫽Young’s modulus of elasticity, psi

G⫽shear modulus of elasticity, psi

␥ ⫽weight density of member, lb / in3

g⫽gravitational acceleration, in / s2

P⫽axial force (tension is positive), lb

A⫽area of member cross section, in2

Cw⫽warping constant

J⫽torsion constant

Ip⫽polar moment of inertia, in4

In design of a truss member, the frequency of vortex shedding for the section is set equalto the bending and torsional frequency and the resulting equation is solved for the wind

speed V This is the wind speed at which resonance occurs The design should be such thatV exceeds by a reasonable margin the velocity at which the wind is expected to occur

13.9TRUSS MEMBER DETAILS

The following shapes for truss members are typically considered:

H sections, made with two side segments (composed of angles or plates) with solid web,

perforated web, or web of stay plates and lacing Modern bridges almost exclusively useH sections made of three plates welded together.

Channel sections, made with two angle segments, with solid web, perforated web, or

web of stay plates and lacing These are seldom used on modern bridges.

Single box sections, made with side channels, beams, angles and plates, or side segments

of plates only The side elements may be connected top and bottom with solid plates,perforated plates, or stay plates and lacing Alternatively, they may be connected at thetop with solid cover plates and at the bottom with perforated plates, or stay plates andlacing Modern bridges use primarily four-plate welded box members The cover platesare usually solid, except for access holes for bolting joints.

Double box sections, made with side channels, beams, angles and plates, or side segments

of plates only The side elements may be connected together with top and bottom forated cover plates, or stay plates and lacing.

per-To obtain economy in member design, it is important to vary the area of steel in ance with variations in total loads on the members The variation in cross section plus theuse of appropriate-strength grades of steel permit designers to use essentially the weight ofsteel actually required for the load on each panel, thus assuring an economical design.

accord-With respect to shop fabrication of welded members, the H shape usually is the mosteconomical section It requires four fillet welds and no expensive edge preparation Require-ments for elimination of vortex shedding, however, may offset some of the inherent economyof this shape.

Box shapes generally offer greater resistance to vibration due to wind, to buckling incompression, and to torsion, but require greater care in selection of welding details Forexample, various types of welded cover-plate details for boxes considered in design of thesecond Greater New Orleans Bridge and reviewed with several fabricators resulted in theobservations in Table 13.5.

Additional welds placed inside a box member for development of the cover plate withinthe connection to the gusset plate are classified as AASHTO category E at the terminationof the inside welds and should be not be used For development of the cover plate withinthe gusset-plate connection, groove welds, large fillet welds, large gusset plates, or a com-bination of the last two should be used.

Tension Members. Where practical, these should be arranged so that there will be nobending in the members from eccentricity of the connections If this is possible, then thetotal stress can be considered uniform across the entire net area of the member At a joint,the greatest practical proportion of the member surface area should be connected to thegusset or other splice material.

Designers have a choice of a large variety of sections suitable for tension members,although box and H-shaped members are typically used The choice will be influenced bythe proposed type of fabrication and range of areas required for tension members The designshould be adjusted to take full advantage of the selected type For example, welded platesare economical for tubular or box-shaped members Structural tubing is available with almost22 in2 of cross-sectional area and might be advantageous in welded trusses of moderatespans For longer spans, box-shape members can be shop-fabricated with almost unlimitedareas.

Tension members for bolted trusses involve additional considerations For example, only50% of the unconnected leg of an angle or tee is commonly considered effective, becauseof the eccentricity of the connection to the gusset plate at each end.

To minimize the loss of section for fastener holes and to connect into as large a proportionof the member surface area as practical, it is desirable to use a staggered fastener pattern.In Fig 13.7, which shows a plate with staggered holes, the net width along Chain 1-1 equals

plate width W, minus three hole diameters The net width along Chain 2-2 equals W, minusfive hole diameters, plus the quantity S2/ 4g for each off four gages, where S is the pitchand g the gage.

TABLE 13.5 Various Welded Cover-Plate Designs for Second Greater New Orleans Bridge

Conventional detail Has been used extensively in the past It may besusceptible to lamellar tearing under lateral or torsional loads.

Overlap increases for thicker web plate Cover plate tends to curve up afterwelding.

Very difficult to hold out-to-out dimension of webs due to thickness toleranceof the web plates Groove weld is expensive, but easier to develop coverplate within the connection to gusset plate.

The detail requires a wide cover plate and tight tolerance of the cover-platewidth With a large overlap, the cover may curve up after welding Grooveweld is expensive, but easier to develop cover plate within the connectionto the gusset plate.

Same as above, except the fabrication tolerance, which will be better withthis detail.

FIGURE 13.7 Chains of bolt holes used for determining thenet section of a tension member.

Compression Members. These should be arranged to avoid bending in the member fromeccentricity of connections Though the members may contain fastener holes, the gross areamay be used in design of such columns, on the assumption that the body of the fastener fillsthe hole Welded box and H-shaped members are typically used for compression membersin trusses.

Compression members should be so designed that the main elements of the section areconnected directly to gusset plates, pins, or other members It is desirable that membercomponents be connected by solid webs Care should be taken to ensure that the criteria forslenderness ratios, plate buckling, and fastener spacing are satisfied.

Posts and Hangers. These are the vertical members in truss bridges A post in a Warrendeck truss delivers the load from the floorbeam to the lower chord A hanger in a Warrenthrough-truss delivers the floorbeam load to the upper chord.

Posts are designed as compression members The posts in a single-truss span are generallymade identical At joints, overall dimensions of posts have to be compatible with those ofthe top and bottom chords to make a proper connection at the joint.

Hangers are designed as tension members Although wire ropes or steel rods could beused, they would be objectionable for esthetic reasons Furthermore, to provide a slendernessratio small enough to maintain wind vibration within acceptable limits will generally requirerope or rod area larger than that needed for strength.

Truss-Member Connections. Main truss members should be connected with gusset platesand other splice material, although pinned joints may be used where the size of a boltedjoint would be prohibitive To avoid eccentricity, fasteners connecting each member shouldbe symmetrical about the axis of the member It is desirable that fasteners develop the fullcapacity of each element of the member Thickness of a gusset plate should be adequate forresisting shear, direct stress, and flexure at critical sections where these stresses are maxi-mum Re-entrant cuts should be avoided; however, curves made for appearance are permis-sible.

13.10MEMBER AND JOINT DESIGN EXAMPLES—LFD AND SLD

Design of a truss member by the AASHTO LFD and SLD Specifications is illustrated in thefollowing examples, The design includes a connection in a Warren truss in which splicingof a truss chord occurs within a joint Some designers prefer to have the chord run contin-uously through the joint and be spliced adjacent to the joint Satisfactory designs can beproduced using either approach Chords of trusses that do not have a diagonal framing intoeach joint, such as a Warren truss, are usually continuous through joints with a post orhanger Thus, many of the chord members are usually two panels long Because of limitationson plate size and length for shipping, handling, or fabrication, it is sometimes necessary,however, to splice the plates within the length of a member Where this is necessary, commonpractice is to offset the splices in the plates so that only one plate is spliced at any crosssection.

13.10.1Load-Factor Design of Truss Chord

A chord of a truss is to be designed to withstand a factored compression load of 7,878 kipsand a factored tensile load of 1,748 kips Corresponding service loads are 4,422 kips com-pression and 391 kips tension The structural steel is to have a specified minimum yield

stress of 36 ksi The member is 46 ft long and the slenderness factor K is to be taken as

FIGURE 13.8 Cross section of a truss chord with a box section.

unity A preliminary design yields the cross section shown in Fig 13.8 The section has thefollowing properties:

w⫽weight per linear foot⫽0.98 kips

Ten 11⁄4-in-dia bolt holes are provided in each web at the section for the connections atjoints The welds joining the cover plates and webs are minimum size,3⁄8in, and are clas-sified as AASHTO fatigue category B.

Although the AASHTO LFD Specification specifies a load factor for dead load of 1.30,the following computation uses 1.50 to allow for about 15% additional weight due to paint,diaphragms, weld metal and fasteners.

Compression in Chord from Factored Loads. The uniform stress on the section isƒc⫽7878 / 281⫽28.04 ksi

The radius of gyration with respect to the weak axis is

ry⫽ 兹I / Agyg⫽ 兹69,520 / 281⫽15.73 inand the slenderness ratio with respect to that axis is

MDL⫽0.98(46) ⫻12 ⫻1.50 / 10⫽3733 kip-inThe section modulus is

Sg⫽I / cgx ⫽97,770 / (54 / 2⫹0.875)⫽3507 inHence, the maximum compressive bending stress is

ƒb⫽MDL/ Sg⫽3733 / 3507⫽1.06 ksiThe plastic section modulus is

Mp⫽full plastic moment, kip-in, of the section

Z⫽plastic modulus⫽1.31Sg

C⫽equivalent moment factor, taken as 0.85 in this case

Fe⫽Euler buckling stress, ksi, with 0.85 factor⫽0.85E␲2/ (KL / rx)2

The effective length factor K is taken equal to unity and the radius of gyration rxwith respect

to the x axis, the axis of bending, is

The sum of the ratios, 0.981, governs (stability) and is satisfactory The section is satisfactoryfor compression.

Local Buckling. The AASHTO specifications limit the depth-thickness ratio of the websto a maximum of

d / t⫽180 /兹ƒc⫽180 /兹28.04⫽34.0

The actual d / t is 54 / 2.0625⫽26.2⬍ 34.0—OK

Maximum permissible width-thickness ratio for the cover plates is

b / t⫽213.4 /兹ƒc⫽213.4 /兹28.04⫽40.3

The actual b / t is 33.125 / 0.875⫽37.9⬍ 40.3—OK

Tension in Chord from Factored Loads. The following treatment is based on a compositeof AASHTO SLD Specifications for the capacity of tension members, and other aspects fromthe AASHTO LFD Specifications This is done because the AASHTO LFD Specificationshave not been updated Clearly, this is not in complete compliance with the AASHTO LFDSpecifications Based on the above, the tensile capacity will be the lesser of the yield strengthtimes the design gross area, or 90% of the tensile strength times the net area Both areas aredefined below For determinations of the design strength of the section, the effect of the boltholes must be taken into account by deducting the area of the holes from the gross sectionarea to obtain the net section area Furthermore, the full gross area should not be used if theholes occupy more than 15% of the gross area When they do, the excess above 15% of theholes not greater than 1-1⁄4in in diameter, and all of area of larger holes, should be deductedfrom the gross area to obtain the design gross area The holes occupy 10⫻ 1.25 ⫽12.50in of web-plate length, and 15% of the 54-in plate is 8.10 in The excess is 4.40 in Hence,

the net area is An⫽ 281⫺ 12.50 ⫻ 2.0625⫽ 255 in2 and the design gross area, ADG⫽

281⫺2⫻4.40⫻2.0625⫽263 in2 The tensile capacity is the lesser of 0.90⫻255⫻58

⫽13,311 kips or 263⫻36 ⫽9,468 kips Thus, the design gross section capacity controlsand the tensile capacity is 9,468 kips.

For computation of design gross moment of inertia, assume that the excess is due to 4

bolts, located 7 and 14 in on both sides of the neutral axis in bending about the x axis.

Equivalent diameter of each hole is 4.40 / 4⫽1.10 in The deduction from the gross moment